Generalising the scattered property of subspaces

Generalising the scattered property of subspaces

Bence Csajb´ok, Giuseppe Marino, Olga Polverino and Ferdinando Zullo ^∗ July 25, 2020

Abstract

LetV be anr-dimensionalF^qn-vector space. We call anF^q-subspace U of V h-scattered if U meets the h-dimensional Fqⁿ-subspaces of V in Fq-subspaces of dimension at most h. In 2000 Blokhuis and Lavrauw proved that dim_FqU ≤ rn/2 when U is 1-scattered. Sub- spaces attaining this bound have been investigated intensively because of their relations with projective two-weight codes and strongly regular graphs. MRD-codes with a maximum idealiser have also been linked to rn/2-dimensional 1-scattered subspaces and ton-dimensional (r−1)- scattered subspaces.

In this paper we prove the upper boundrn/(h+1) for the dimension of h-scattered subspaces, h > 1, and construct examples with this dimension. We study their intersection numbers with hyperplanes, introduce a duality relation among them, and study the equivalence problem of the corresponding linear sets.

1 Introduction

LetV(n, q) denote ann-dimensionalFq-vector space. At-spread ofV(n, q) is a setS oft-dimensionalFq-subspaces such that each vector ofV(n, q)\{0}is contained in exactly one element ofS. As shown by Segre in [26], at-spread ofV(n, q) exists if and only if t|n.

LetV be an r-dimensionalFqⁿ-vector space and let S be an n-spread of V, viewed as anFq-vector space. AnFq-subspace U of V is calledscattered

∗The research was supported by the Italian National Group for Algebraic and Geomet- ric Structures and their Applications (GNSAGA - INdAM). The first author was partially supported by the J´anos Bolyai Research Scholarship of the Hungarian Academy of Sciences and by OTKA grants PD 132463 and K 124950. The last two authors were supported by the project “VALERE: VAnviteLli pEr la RicErca” of the University of Campania “Luigi Vanvitelli”.

w.r.t. S if it meets every element of S in an Fq-subspace of dimension at most one, see [4]. If we consider V as an rn-dimensional Fq-vector space, then it is well-known that the one-dimensional Fqⁿ-subspaces of V, viewed asn-dimensionalFq-subspaces, form ann-spread ofV. This spread is called theDesarguesian spread. In this paper scattered will always mean scattered w.r.t. the Desarguesian spread. For such subspaces Blokhuis and Lavrauw showed that their dimension can be bounded by rn/2. After a series of papers, it is now known that when 2|rn then there always exist scattered subspaces of this dimension [1, 3, 4, 11].

In this paper we introduce and study the following special class of scat- tered subspaces.

Definition 1.1. LetV be anr-dimensionalFqⁿ-vector space. AnFq-subspace U of V is called h-scattered, 0 < h ≤ r−1, if hUi_Fqn = V and each h- dimensional Fqⁿ-subspace of V meets U in an Fq-subspace of dimension at most h. An h-scattered subspace of highest possible dimension is called a maximumh-scattered subspace.

With this definition, the 1-scattered subspaces are the scattered sub- spaces generating V over Fqⁿ. With h =r the above definition would give then-dimensionalFq-subspaces ofV defining subgeometries of PG(V,Fqⁿ).

Ifh=r−1 and dim_FqU =n, then U defines a scattered Fq-linear set with respect to hyperplanes, introduced in [28, Definition 14]. A further gener- alisation of the concept ofh-scattered subspaces can be found in the recent paper [2].

In this paper we prove that for an h-scattered subspaceU ofV(r, qⁿ), if U does not define a subgeometry, then

dim_FqU ≤ rn

h+ 1, (1)

cf. Theorem 2.3. Clearly,h-scattered subspaces reaching bound (1) are max- imum h-scattered. When h+ 1|r then our examples prove that maximum h-scattered subspaces have dimensionrn/(h+ 1), cf. Theorem 2.6. In The- orem 2.7 we show thath-scattered subspaces of dimensionrn/(h+ 1) meet hyperplanes ofV(r, qⁿ) inFq-subspaces of dimension at leastrn/(h+ 1)−n and at most rn/(h+ 1)−n+h. Then we introduce a duality relation be- tween maximum h-scattered subspaces of V(r, qⁿ) reaching bound (1) and maximum (n−h−2)-scattered subspaces ofV(rn/(h+ 1)−r, qⁿ) reaching bound (1), which allows us to give some constructions also when h+ 1 is not a divisor ofr, cf. Theorem 3.6.

Proposition 2.1 shows us thath-scattered subspaces are special classes of 1-scattered subspaces. In [28, Corollary 4.4] the (r−1)-scattered subspaces ofV(r, qⁿ) attaining bound (1), i.e. of dimensionn, have been shown to be equivalent to MRD-codes of F^n×n_q with minimum rank distance n−r+ 1 and with left or right idealiser isomorphic to Fqⁿ. In Section 4 we study theFq-linear set L_U determined by anh-scattered subspaceU. In contrast to the case of 1-scattered subspaces, it turns out that for any h-scattered Fq-subspacesU and W ofV(r, qⁿ) withh >1, the corresponding linear sets L_U andL_W are PΓL(r, qⁿ)-equivalent if and only ifU and W are ΓL(r, qⁿ)- equivalent, cf. Theorem 4.5. Forr > 2 this result extends [28, Proposition 3.5] regarding the equivalence between MRD-codes and maximum (r−1)- scattered subspaces attaining bound (1) into an equivalence between MRD- codes and the corresponding linear sets, see [28, Remarks 4, 5].

2 The maximum dimension of an h-scattered sub- space

We start this section by the following result.

Proposition 2.1. For h > 1 the h-scattered subspaces are also i-scattered for anyi < h. In particular they are all 1-scattered.

Proof. LetU be anh-scattered subspace ofV. Suppose to the contrary that it is noti-scattered for somei < h. Therefore, there exists ani-dimensional Fqⁿ-subspace S such that dim_Fq(S∩U)≥i+ 1. As hUi_Fqn =V, there exist u₁, . . . ,u_h−i ∈U such that dim_FqnhS,u₁, . . . ,u_h−ii_Fqn =h. Then

dim_Fq U∩ hS,u1, . . . ,uh−ii_Fqn

≥(i+ 1) + (h−i) =h+ 1, a contradiction.

In the proof of the main result of this section we will need the following lemma.

Lemma 2.2. For any integer iwith r≤i≤n in V =V(r, qⁿ) there exists an(r−1)-scattered Fq-subspace of dimensioni.

Proof. Fix anFqⁿ-basis ofV, then the spaceV can be seen asF^r_qⁿ. Consider then-dimensional Fq-subspace U ={(x, x^q, . . . , x^qr−1) :x∈Fqⁿ}of V. Let

W be any i-dimensional Fq-subspace of U. The intersection of W with a hyperplane [a₀, a₁, . . . , ar−1] of V is







(x, x^q, . . . , x^qr−1) :x∈Fqⁿ,

r−1

X

j=0

ajx^qj = 0







∩W,

which is clearly an Fq-subspace of size at most degPr−1

j=0ajx^qj ≤ q^r−1. If hWi_Fqn 6= V then there was a hyperplane of V containing W, a contradic- tion, i.e. W is an (r−1)-scatteredFq-subspace ofV.

Forh= 1, the following result was shown in [4].

Theorem 2.3. Let V be an r-dimensional Fqⁿ-vector space and U an h- scattered Fq-subspace ofV. Then either

• dim_FqU =r, U defines a subgeometry of PG(V,Fqⁿ)and U is(r−1)- scattered, or

• dim_FqU ≤rn/(h+ 1).

Proof. Letkdenote the dimension ofU overFq. SincehUi_Fqn =V, we have k≥rand in case of equalityU defines a subgeometry of PG(V,Fqⁿ) which is clearly (r−1)-scattered. From now on we may assumek > r. First consider the case h = r−1. Fix an Fqⁿ-basis in V and for x ∈ V denote the i-th coordinate w.r.t. this basis by xi. Consider the following set of Fq-linear maps fromU toFqⁿ:

C_U :=

(

Ga0,...,ar−1:x∈U 7→

r−1

X

i=0

aixi:ai ∈Fqⁿ

) .

First we show that the non-zero maps ofC_U have rank at leastk−r+ 1. In- deed, if (a₀, . . . , ar−1)6=0, thenu∈kerG_{a0,...,ar−1} if and only ifPr−1

i=0a_iu_i = 0, i.e. kerGa0,...,ar−1 =U∩H, whereH is the hyperplane [a0, a1, . . . , ar−1] of V. SinceU is (r−1)-scattered, it follows that dim_FqkerG_{a0,...,ar−1} ≤r−1 and hence the rank ofG_{a0,...,ar−1} is at leastk−r+ 1. Next we show that any two maps ofC_U are different. Suppose to the contrary Ga0,...,ar−1 =Gb0,...,br−1, thenG_a0−b₀,...,ar−1−br−1 is the zero map. If (a₁−b₁, . . . , a_r−b_r) 6=0, then U would be contained in the hyperplane [a₀−b₀, a₁−b₁, . . . , ar−1−br−1], a contradiction sincehUi_Fqn =V. Hence,|C_U|=q^nr.

Suppose to the contraryk > n. The elements ofC_Uform anr-dimensional Fq-subspace of Hom_Fq(U,Fqⁿ) and the non-zero maps of C_U have rank at

leastk−r+ 1. By Result 4.6 (Singleton-like bound) we getq^rn≤q^k(n−k+r) and hence (k−n)(k−r)≤0, which contradictsk > r.

From now on, we will assume 1< h < r−1, since the assertion has been proved in [4] forh= 1.

First we assume n≥h+ 1. Then by Lemma 2.2, in F^hqⁿ there exists an (h−1)-scattered Fq-subspace W of dimensionh+ 1.

Let G be an Fq-linear transformation from V to itself with kerG =U. Clearly, dim_FqImG = rn−k. For each (u₁, . . . ,u_h) ∈ V^h consider the Fqⁿ-linear map

τ_u1,...,uh: (λ₁, . . . , λ_h)∈W 7→λ₁u₁+. . .+λ_hu_h∈V.

Consider the following set ofFq-linear maps W →ImG C :={G◦τu1,...,uh : (u1, . . . ,uh)∈V^h}.

Our aim is to show that these maps are pairwise distinct and hence |C| = q^rnh. Suppose G◦τu1,...,uh=G◦τv1,...,vh. It follows thatG◦τu1−v1,...,uh−vh

is the zero map, i.e.

λ1(u1−v1) +. . .+λh(uh−vh)∈kerG=U for each (λ1, . . . , λh)∈W. (2) For i ∈ {1, . . . , h}, put zi = ui −vi, let T := hz₁, . . . ,zhi_qⁿ and let t = dimqⁿT. We want to show thatt= 0. If t=h, then by (2)

{λ₁z1+. . .+λhzh : (λ1, . . . , λh)∈W} ⊆T∩U,

hence dim_Fq(T∩U)≥dim_FqW =h+ 1, which is not possible sinceT is an h-dimensional Fqⁿ-subspace of V and U is h-scattered. Hence 0 ≤ t < h.

Assumet≥1. Let Φ :F^hqⁿ →T be the Fqⁿ-linear map defined by the rule (λ1, . . . , λ_h)7→λ1z1+. . .+λ_hz_h

and consider the mapτz1,...,zh. Note that τz1,...,zh is the restriction of Φ on theFq-vector subspaceW ofF^hqⁿ. It can be easily seen that

dim_Fqnker Φ =h−t, (3)

kerτ_z1,...,zh = ker Φ∩W, (4)

and by (2)

Imτz1,...,zh ⊆T∩U. (5)

Sincet≥1, by Proposition 2.1 theFq-subspaceW is (h−t)-scattered inF^hqⁿ

and hence taking (3) and (4) into account we get dim_Fqkerτ_z1,...,zh≤h−t, which yields

dim_FqImτz1,...,zh ≥t+ 1. (6)

By Proposition 2.1 the Fq-subspace U is also a t-scattered subspace of V, thus by (5)

dim_FqImτ_z1,...,zh ≤dim_Fq(T∩U)≤t,

contradicting (6). It follows that t = 0, i.e. z_i = 0 for each i∈ {1, . . . h}

and hence | C |=q^rnh. The trivial upper bound for the size of C is the size ofF(h+1)×(rn−k)

q , thus

q^rnh =| C | ≤q(h+1)(rn−k), which implies

k≤ rn h+ 1.

Now assume n < h + 1. By Proposition 2.1 U is h⁰-scattered with h⁰ = n−1. Since h⁰ < r−1 and n ≥ h⁰ + 1, we can argue as before and derivek= dim_FqU ≤rn/(h⁰+ 1) =r, contradictingk > r.

The previous proof can be adapted also for the h = 1 case without introducing the subspace W, cf. [30].

The following result is a generalisation of [3, Theorem 3.1].

Theorem 2.4. LetV =V1⊕. . .⊕V_twhereVi=V(ri, qⁿ) andV =V(r, qⁿ).

If U_i is an h_i-scattered Fq-subspace in V_i, then the Fq-subspace U = U₁⊕ . . .⊕Ut is h-scattered in V, with h = min{h₁, . . . , ht}. Also, if Ui is h- scattered in Vi and its dimension reaches bound (1), then U is h-scattered in V and its dimension reaches bound (1).

Proof. Clearly, it is enough to prove the assertion fort= 2.

If h = 1, the result easily follows from Proposition 2.1 and from [3, Theorem 3.1]; hence, we may assumeh=h₁ ≥2.

By way of contradiction suppose that there exists anh-dimensionalFqⁿ- subspaceW of V such that

dim_Fq(W ∩U)≥h+ 1. (7)

Clearly, W cannot be contained in V₁ since U₁ is h-scattered in V₁. Let W₁ := W ∩V₁ and s := dim_FqnW₁. Then s < h and by Proposition 2.1,

theFq-subspace U1 iss-scattered inV1, thus dim_Fq(U1∩W1)≤s. Denoting hU₁, W ∩Ui_Fq by ¯U₁, the Grassmann formula and (7) yield

dim_FqU¯1−dim_FqU1≥h+ 1−s. (8) Consider the subspaceT :=W +V1 of the quotient spaceV /V1 ∼=V2. Then dim_Fqn T =h−sand T contains theFq-subspace

M := ¯U1+V1.

Since M is also contained in theFq-subspace U +V₁ =U₂+V₁, thenM is h2-scattered in V /V1 and hence by h−s≤h ≤h2 and by Proposition 2.1, M is also (h−s)-scattered inV /V₁.

On the other hand,

dim_Fq(M ∩T) = dim_FqM = dim_FqU¯₁−dim_Fq( ¯U₁∩V₁)≥ dim_FqU¯1−dim_Fq(U ∩V1) = dim_FqU¯1−dim_FqU1, and hence, by (8),

dim_Fq(M∩T)≥h−s+ 1, a contradiction.

The last part follows from rn/(h+ 1) =Pt

i=1rin/(h+ 1).

Constructions of maximum 1-scatteredFq-subspaces ofV(r, qⁿ) exist for all values ofq,r andn, providedrnis even [1, 3, 4, 11]. Forr= 3,n≤5 see [2, Section 5]. Also, there are constructions of maximum (r−1)-scattered Fq-subspaces arising from MRD-codes (explained later in Section 4.1) for all values of q, r and n, cf. [28, Corollary 4.4]. In particular, the so called Gabidulin codes produce Example 2.5. One can also prove directly that these are maximum (r−1)-scattered subspaces by the same arguments as in the proof of Lemma 2.2.

Example 2.5. In F^r_qⁿ, if n≥r, then the Fq-subspace

{(x, x^q, x^q2, . . . , x^qr−1) :x∈Fqⁿ} is maximum (r−1)-scattered of dimension n.

Theorem 2.6. If h+ 1 dividesr andn≥h+ 1, then inV =V(r, qⁿ) there exist maximum h-scattered Fq-subspaces of dimension rn/(h+ 1).

Proof. Put r = t(h+ 1) and consider V = V1⊕. . .⊕Vt, with Vi an Fqⁿ- subspace of V with dimension h+ 1. For each i consider a maximum h- scattered Fq-subspace U_i in V_i of dimension n which exists because of Ex- ample 2.5. By Theorem 2.4,U1⊕. . .⊕Ut is anh-scattered Fq-subspace of V with dimension tn= _h+1^rn .

In Theorem 2.6 we exhibit examples of maximum h-scattered subspaces ofV =V(r, qⁿ) wheneverh+1 dividesr. In Section 3 we introduce a method to construct such subspaces also whenh+1 does not divider. To do this, we will need an upper bound on the dimension of intersections of hyperplanes of V with a maximum h-scattered subspace of dimension rn/(h+ 1). The proof of the following theorem is developed in Section 5.

Theorem 2.7. If U is a maximumh-scatteredFq-subspace of a vector space V(r, qⁿ) of dimension rn/(h+ 1), then for any (r −1)-dimensional Fqⁿ- subspaceW of V(r, qⁿ) we have

rn

(h+ 1)−n≤dim_Fq(U ∩W)≤ rn

(h+ 1) −n+h.

The above theorem is a generalisation of [4, Theorem 4.2] and the first part of its proof relies on the counting technique developed in [4, Theorem 4.2].

3 Delsarte dual of an h-scattered subspace

LetU be ak-dimensional Fq-subspace of a vector space Λ =V(r, qⁿ), with k > r. By [21, Theorems 1, 2] (see also [20, Theorem 1]), there is an embedding of Λ inV=V(k, qⁿ) withV= Λ⊕Γ for some (k−r)-dimensional Fqⁿ-subspace Γ such that U = hW,Γi_Fq ∩Λ, where W is a k-dimensional Fq-subspace ofV, hWi_Fqn =V and W ∩Γ ={0}. Then the quotient space V/Γ is isomorphic to Λ and under this isomorphism U is the image of the Fq-subspaceW + Γ ofV/Γ.

Now, let β⁰:W ×W → Fq be a non-degenerate reflexive sesquilinear form on W with companion automorphism σ⁰. Then β⁰ can be extended to a non-degenerate reflexive sesquilinear form β: V×V→ Fqⁿ. Indeed if {u₁, . . . ,u_k} is an Fq-basis of W, since hWi_Fqn =V, for each v,w ∈V we have

β(v,w) =

k

X

i,j=1

a_ib^σ_jβ⁰(u_i,u_j),

wherev=Pk

i=1aiui,w=Pk

i=1biui andσ is an automorphism ofFqⁿ such that σ_|Fq =σ⁰. Let ⊥and ⊥⁰ be the orthogonal complement maps defined byβ and β⁰ on the lattice of Fqⁿ-subspaces of Vand of Fq-subspaces ofW, respectively. For anFq-subspace S of W the Fqⁿ-subspace hSi_Fqn of Vwill be denoted by S^∗. In this case (S^∗)^⊥= (S^⊥0)^∗.

In this setting, we can prove the following preliminary result.

Proposition 3.1. Let W, Λ, Γ,V, ⊥and⊥⁰ be defined as above. If U is a k-dimensional Fq-subspace ofΛ withk > r and

dim_Fq(M∩U)< k−1 holds for each hyperplaneM of Λ, () thenW+Γ^⊥is ak-dimensionalFq-subspace of the quotient spaceV/Γ^⊥. Proof. As described above,U turns out to be isomorphic to theFq-subspace W + Γ of the quotient space V/Γ. By (), since each hyperplane ofV/Γ is of formH+ Γ whereH is a hyperplane of Vcontaining Γ, it follows that

dim_Fq(H∩W)< k−1 for each hyperplaneH ofVcontaining Γ. ()

To prove the assertion it is enough to prove W ∩Γ^⊥ ={0}.

Indeed, by way of contradiction, suppose that there exists a nonzero vector v∈W ∩Γ^⊥. Then theFqⁿ-hyperplane hvi^⊥

Fqn ofVcontains the subspace Γ and meetsW in the (k−1)-dimensionalFq-subspacehvi^⊥0

F^q, which contradicts ().

Definition 3.2. LetU be ak-dimensionalFq-subspace of Λ =V(r, qⁿ), with k > r and such that () is satisfied. Then the k-dimensional Fq-subspace W + Γ^⊥ of the quotient space V/Γ^⊥ (cf. Proposition 3.1) will be denoted by ¯U and we call it theDelsarte dual of U (w.r.t.⊥).

The term Delsarte dual comes from the Delsarte dual operation acting on MRD-codes, as pointed out in Theorem 4.12.

Theorem 3.3. Let U be a maximum h-scattered Fq-subspace of a vector space Λ = V(r, qⁿ) of dimension rn/(h + 1), with n ≥ h+ 3. Then the Fq-subspace U¯ of V/Γ^⊥ = V(rn/(h+ 1)−r, qⁿ) obtained by the procedure of Proposition 3.1 is maximum (n−h−2)-scattered.

Proof. Put k := rn/(h+ 1). We first note that condition () is satisfied for U since by Theorem 2.7 the hyperplanes of Λ meet U in Fq-subspaces of dimension at most rn/(h+ 1)−n+h < k−1. Also, k > r holds since n≥h+ 3.

Hence we can apply the procedure of Proposition 3.1 to obtain the Fq- subspace ¯U =W + Γ^⊥ of V/Γ^⊥ of dimensionk.

By way of contradiction, suppose that there exists an (n− h − 2)- dimensionalFqⁿ-subspace of V/Γ^⊥, say M, such that

dim_Fq(M∩U¯)≥n−h−1. (9) Then M =H+ Γ^⊥, for some (n+r−h−2)-dimensional Fqⁿ-subspace H ofV containing Γ^⊥. ForH, by (9), it follows that

dim_Fq(H∩W) = dim_Fq(M∩U¯)≥n−h−1.

LetS be an (n−h−1)-dimensionalFq-subspace of W contained inH and letS^∗ :=hSi_Fqn. Then, dim_FqnS^∗ =n−h−1,

S^⊥0 =W ∩(S^∗)^⊥ and S^⊥0 ⊂(S^∗)^⊥=hS^⊥0i_Fqn. (10) SinceS ⊆H∩W and Γ^⊥⊂H, we getS^∗ ⊂H andH^⊥ ⊂Γ, i.e.

H^⊥⊆Γ∩(S^∗)^⊥. (11)

From (11) it follows that dim_Fqn

Γ∩(S^∗)^⊥

≥dim_Fqn H^⊥=k−(n+r−h−2).

This implies that

dim_FqnhΓ,(S^∗)^⊥i_Fqn = dim_FqnΓ+dim_Fqn(S^∗)^⊥−dim_Fqn

Γ∩(S^∗)^⊥

≤k−1

and hence hΓ,(S^∗)^⊥i_Fqn is contained in a hyperplane T of V containing Γ.

Also, dim_Fq(S^⊥0) = dim_FqW −dim_FqS =k−(n−h−1) and, by (10), we get

S^⊥0 =W ∩(S^∗)^⊥⊆W ∩T.

Then ˆT :=T∩Λ is a hyperplane of Λ and, by recallingU =hW,Γi_Fq ∩Λ, dim_Fq( ˆT∩U) = dim_Fq(T∩W)≥dim_Fq(S^⊥0) =k−n+h+ 1, contradicting Theorem 2.7.

In case ofh=r−1, Theorem 3.3 follows from [28] and from the theory of MRD codes. Our theorem generalises this result to each value of h by using a geometric approach.

Corollary 3.4. Starting from a maximum (r−1)-scattered Fq-subspace U ofV(r, qⁿ) of dimensionn,n≥r+ 2, theFq-subspaceU¯ (cf. Definition 3.2) is a maximum(n−r−1)-scatteredFq-subspace ofV(n−r, qⁿ)of dimension n.

Corollary 3.5. Starting from a maximum 1-scattered Fq-subspace U of V(r, qⁿ), rn even, n ≥ 4, U¯ (cf. Definition 3.2) is a maximum (n−3)- scattered Fq-subspace of V(r(n−2)/2, qⁿ) whose dimension attains bound (1).

Theorem 3.6. Ifn≥4is even andr≥3 is odd, then there exist maximum (n−3)-scattered Fq-subspaces of V(r(n−2)/2, qⁿ) which cannot be obtained from the direct sum construction of Theorem 2.6.

Proof. By [1, 3, 4, 11] it is always possible to construct maximum 1-scattered Fq-subspaces of V(r, qⁿ). Then the result follows from Corollary 3.5 and from the fact that in this casen−2 does not divider(n−2)/2.

Remark 3.7. The Delsarte dual of anFq-subspace does not depend on the choice of the non-degenerate reflexive sesquilinear form onW.

Indeed, fix anFq-basis B of W, sincehWi_Fqn =V, we can seeW asF^k_q and V as F^kqⁿ. Let β₁⁰ and β⁰₂ be two non-degenerate reflexive sesquilinear forms on F^kq. Then, with respect to the basis B, the forms β⁰₁ and β₂⁰ are defined by the following rules:

β_i⁰((x,y)) =xGiy^ρ_t^{i 1},

whereG_i ∈GL(k, q) andρ_i is an automorphism ofFqsuch thatρ²_i = id and (G^ρ_iⁱ)^t =Gi, for i∈ {1,2}. Now letβ1 and β2 be their extensions overF^k_qⁿ defined by the rules

β_i((x,y)) =xG_iy^ρ_tⁱ,

and let ⊥₁ and ⊥₂ be the orthogonal complement maps defined by β1 and β2 on the lattice ofFqⁿ-subspaces ofF^kqⁿ, respectively.

Again w.r.t. the basisB, the Fqⁿ-subspace Γ described at the beginning of this section can be seen as a (k−r)-dimensional subspace ofF^k_qⁿ. Then, fori∈ {1,2} we have

Γ^⊥i ={x:xG_iy^ρ_tⁱ = 0 ∀y∈Γ}.

1Hereytdenotes the transpose of the vectory.

Straightforward computations show that the invertible semilinear map ϕ:x∈F^k_qⁿ 7→x^{ρ−12 ρ1}G^ρ

−1 2 ρ1

2 G⁻¹₁ ∈F^k_qⁿ,

leavesW invariant and maps Γ^⊥2 to Γ^⊥1. ThenϕmapsW+Γ^⊥2 toW+Γ^⊥1, i.e. ϕ maps the Delsarte dual ofU calculated w.r.t β₂ to the Delsarte dual ofU calculated w.r.t. β1. See also [25, Section 2] and [27, Section 6.2].

4 Linear sets defined by h-scattered subspaces

LetV be anr-dimensionalFqⁿ-vector space. A point setLof Λ = PG(V,Fqⁿ)

= PG(r−1, qⁿ) is said to be an Fq-linear set of Λ of rank kif it is defined by the non-zero vectors of ak-dimensionalFq-vector subspaceU ofV, i.e.

L=L_U :={hui_Fqn :u∈U \ {0}}.

One of the most natural questions about linear sets is their equivalence.

Two linear setsL_UandL_W of PG(r−1, qⁿ) are said to be PΓL-equivalent (or simplyequivalent) if there is an elementϕin PΓL(r, qⁿ) such thatL^ϕ_U =LW. In the applications it is crucial to have methods to decide whether two linear sets are equivalent or not. This can be a difficult problem and some results in this direction can be found in [9, 8, 12]. For f ∈ ΓL(r, qⁿ) we have L_Uf =L^ϕ_U^f, where ϕ_f denotes the collineation of PG(V,Fqⁿ) induced by f. It follows that if U and W are Fq-subspaces of V belonging to the same orbit of ΓL(r, qⁿ), then LU and LW are equivalent. The above condition is only sufficient but not necessary to obtain equivalent linear sets. This follows also from the fact thatFq-subspaces ofV with different dimensions can define the same linear set, for example Fq-linear sets of PG(r −1, qⁿ) of rank k≥rn−n+ 1 are all the same: they coincide with PG(r−1, qⁿ).

Also, in [8, 12] for r = 2 it was pointed out that there exist maximum 1-scatteredFq-subspaces of V on different orbits of ΓL(2, qⁿ) defining PΓL- equivalent linear sets of PG(1, qⁿ). It is then natural to ask for which linear sets can we translate the question of PΓL-equivalence into the question of ΓL-equivalence of the defining subspaces. For further details on linear sets see [17, 18, 24].

In this section we study the equivalence issue ofFq-linear sets defined by h-scattered linear sets forh≥2.

Definition 4.1. If U is a (maximum) h-scattered Fq-subspace of V(r, qⁿ), then theFq-linear setL_U of PG(r−1, qⁿ) is called (maximum)h-scattered.

The (r−1)-scattered Fq-linear sets of rank n were defined also in [28, Definition 14] and following the authors of [28], we will call theseFq-linear sets maximum scattered with respect to hyperplanes. Also, we will call 2- scatteredFq-linear sets (of any rank)scattered with respect to lines.

Proposition 4.2 ([5, pg. 3 Eq. (6) and Lemma 2.1]). Let V be a two- dimensional vector space over Fqⁿ.

1. If U is an Fq-subspace of V with|L_U|=q+ 1, then U has dimension 2 over Fq.

2. Let U and W be two Fq-subspaces of V with L_U =L_W of size q+ 1.

If U∩W 6={0}, then U =W.

Proposition 4.3. If L_U is a scattered Fq-linear set with respect to lines of PG(r−1, qⁿ) = PG(V,Fqⁿ), then its rank is uniquely defined, i.e. for each Fq-subspaceW of V if L_W =L_U, then dim_FqW = dim_FqU.

Proof. Let W be an Fq-subspace of V such that L_U = L_W and put k = dim_FqU. Since U is a 1-scatteredFq-subspace (cf. Proposition 2.1),|L_U|=

|L_W| = (q^k −1)/(q −1). It follows that dim_FqW ≥ k. Suppose that dim_FqW ≥k+ 1, then there exists at least one pointP =hxi_Fqn ∈L_W such that dim_Fq(W∩hxi_Fqn)≥2. LetQ=hyi_Fqn ∈L_U =L_W be a point different fromP, thenhx,yi_Fqn∩W has dimension at least 3 but the linear set defined byhx,yi_Fqn ∩W isL_W ∩ hP, Qi, thus it has size q+ 1, contradicting part 1 of Proposition 4.2.

Lemma 4.4. Let L_U be a scattered Fq-linear set with respect to lines in PG(r −1, qⁿ). If L_U = L_W for some Fq-subspace W, then U = λW for some λ∈F^∗_qⁿ.

Proof. By Proposition 4.3, we have dim_FqW = dim_FqU and hence, since U is 1-scattered, also W is 1-scattered. Let P ∈ L_U with P = hui_Fqn, then for some λ ∈ F^∗qⁿ we have u ∈ U ∩λW. Put W⁰ := λW and note that LW =LW⁰. Our aim is to prove W⁰ ⊆U. SinceU and W⁰ are 1-scattered, we have hui_Fqn ∩U =hui_Fqn ∩W⁰=hui_Fq.

What is left, is to show for each w ∈ W⁰\ hui_Fqn that w ∈ U. To do this, consider the point Q=hwi_Fqn ∈LW⁰ =LU and the line hP, Qi which meets L_U in q+ 1 points. By part 1 of Proposition 4.2, the Fq-subspace

hu,wi_Fqn ∩U

has dimension 2. Since hu,wi_Fqn ∩U

∩ hu,wi_Fqn ∩W⁰ 6=

{0}, by part 2 of Proposition 4.2 we get

hu,wi_Fqn ∩U =hu,wi_Fqn ∩W⁰ =hu,wi_Fq. Hence the assertion follows.

Theorem 4.5. Consider twoh-scattered linear setsLU and LW ofV(r, qⁿ) with h ≥ 2. They are PΓL(r, qⁿ)-equivalent if and only if U and W are ΓL(r, qⁿ)-equivalent.

Proof. The if part is trivial. To prove the only if part assume that there exists f ∈ ΓL(r, qⁿ) such that L^ϕ_U^f = LW, where ϕ_f is the collineation induced byf. Since L^ϕ_U^f =L_Uf, by Proposition 2.1 and Lemma 4.4, there exists λ ∈ F^∗_qⁿ such that λU^f = W and hence U and W lie on the same orbit of ΓL(r, qⁿ).

4.1 Scattered linear sets with respect to hyperplanes and MRD-codes

A rank distance (or RD) code C of F^n×mq , n ≤ m, can be considered as a subset of Hom_Fq(U, V), where dim_FqU = m and dim_FqV = n, with rank distance defined as d(f, g) := rk(f −g). The minimum distance of C is d:= min{d(f, g) :f, g ∈ C, f 6=g}.

Result 4.6 ([13]). If C is a rank distance code of F^n×m_q , n ≤ m, with minimum distance d, then

|C| ≤q^m(n−d+1). (12) Rank distance codes for which (12) holds with equality are calledmaxi- mum rank distance (or MRD) codes.

From now on, we will only consider Fq-linear MRD-codes of F^n×nq , i.e.

those which can be identified withFq-subspaces of End_Fq(Fqⁿ). Since End_Fq(Fqⁿ) is isomorphic to the ring ofq-polynomials overFqⁿ modulox^qn−x, denoted byL_n,q, with addition and composition as operations, we will considerC as an Fq-subspace of L_n,q. Given two Fq-linear MRD codes, C₁ and C₂, they are equivalent if and only if there exist ϕ1, ϕ2 ∈ L_n,q permuting Fqⁿ and ρ∈Aut(Fq) such that

ϕ1◦f^ρ◦ϕ2 ∈ C₂ for all f ∈ C₁, where◦stands for the composition of maps andf^ρ(x) =P

a^ρ_ix^qi forf(x) = Pa_ix^qi. For a rank distance codeCgiven by a set of linearized polynomials, its left and right idealisers can be written as:

L(C) ={ϕ∈ L_n,q :ϕ◦f ∈ C for all f ∈ C}, R(C) ={ϕ∈ L_n,q:f◦ϕ∈ C for allf ∈ C}.

By [19, Section 2.7] and [28] the next result follows. We give a proof of the first part for the sake of completeness.

Result 4.7. C is an Fq-linear MRD-code of L_n,q with minimum distance n−r + 1 and with left-idealiser isomorphic to Fqⁿ if and only if up to equivalence

C =hf₁(x), . . . , fr(x)i_Fqn for some f1, f2, . . . , fr∈ L_n,q and the Fq-subspace

UC={(f₁(x), . . . , f_r(x)) :x∈Fqⁿ} is a maximum(r−1)-scattered Fq-subspace of F^rqⁿ.

Proof. Let T ={ω_a :a∈Fqⁿ}, where for each a∈Fqⁿ, ωa(x) = ax∈ L_n,q and let L denote the left-idealiser of C. Since T and L are Singer cyclic subgroups of GL(Fqⁿ,Fq) and any two such groups are conjugate (cf. [16, pg. 187]) it follows that there exists an invertible q-polynomialg such that g◦L◦g⁻¹ =T. Then for eachh∈ C⁰ :=g⁻¹◦ C it holds thatωa◦h∈ C⁰ for eacha∈Fqⁿ, which proves the first statement. For the second part see [28, Corollary 4.4].

Remark 4.8. The adjoint of a q-polynomial f(x) = Pn−1

i=0 aix^qi, with re- spect to the bilinear form hx, yi:= Tr_qⁿ_/q(xy) (²), is given by

fˆ(x) :=

n−1

X

i=0

a^q_iⁿ⁻ⁱx^qn−i.

If C is a rank distance code given by q-polynomials, then the adjoint code C^> of C is{fˆ:f ∈ C}. The codeC is an MRD if and only if C^> is an MRD and also L(C) ∼=R(C^>), R(C) ∼=L(C^>). Thus Result 4.7 can be translated also to codes with right-idealiser isomorphic toFqⁿ.

The next result follows from [28, Proposition 3.5].

Result 4.9. LetCandC⁰be twoFq-linear MRD-codes ofL_n,qwith minimum distance n−r+ 1and with left-idealisers isomorphic to Fqⁿ. ThenUC and UC⁰ are ΓL(r, qⁿ)-equivalent if and only if C and C⁰ are equivalent.

By Theorem 4.5, for r > 2 we can extend Result 4.9 in the following way.

Theorem 4.10. Let C and C⁰ be two Fq-linear MRD-codes of L_n,q with minimum distance n−r+ 1, r > 2, and with left-idealisers isomorphic to Fqⁿ. Then the linear setsLUC andLU_C0 arePΓL(r, qⁿ)-equivalent if and only if C and C⁰ are equivalent.

2Where Tr_qn/q(x) =x+x^q+. . .+x^qn−1 denotes theF^qn→F^q trace function.

In the following we motivate why we used the term “Delsarte dual”

in Definition 3.2. In particular, we prove that the duality of Section 3 corresponds to the Delsarte duality on MRD-codes when (r−1)-scattered Fq-subspaces ofF^r_qⁿ are considered.

First recall that in terms of linearized polynomials, the Delsarte dual of a rank distance codeCofL_n,qintroduced in [13] and in [14] can be interpreted as follows

C^⊥={f ∈ L_n,q :b(f, g) = 0 ∀g∈ C}, where b(f, g) = Tr_qⁿ_/q

Pn−1 i=0 aibi

for f(x) = Pn−1

i=0 aix^qi and g(x) = Pn−1

i=0 b_ix^qi ∈ L_n,q.

Remark 4.11. Let C be an Fq-linear MRD-code of L_n,q with minimum distance n−r+ 1 and with left-idealiser isomorphic toFqⁿ. By Result 4.7 and by [10, Theorem 2.2], there exist r distinct integers in {0, . . . , n−1}

such that, up to equivalence,

C=hh₀(x), . . . , hr−1(x)i_Fqn, where

hi(x) =x^qti + X

j /∈{t0,...,tr−1}

gi,jx^qj (13)

and gi,j ∈Fqⁿ.

Also, let {s₀, s1, . . . , sn−r−1} := {0, . . . , n−1} \ {t₀, . . . , tr−1}. Then it is easy to see that the Delsarte dual ofC is

C^⊥=hh⁰₀(x), . . . , h⁰_n−r−1(x)i_Fqn, where

h⁰_i(x) =x^qsi − X

j∈{t₀,...,tr−1}

g_j,six^qj. (14) Theorem 4.12. Let C be an Fq-linear MRD-code of L_n,q with minimum distancen−r+ 1and with left-idealiser isomorphic toFqⁿ. Then there exist h₀(x), . . . , hr−1(x), h⁰₀(x), . . . , h⁰_n−r−1(x)∈ L_n,q such that, up to equivalence,

• C =hh₀(x), . . . , hr−1(x)i_Fqn,

• C^⊥=hh⁰₀(x), . . . , h⁰_n−r−1(x)i_Fqn,

• the Delsarte dual of UC = {(h₀(x), . . . , hr−1(x)) : x ∈ Fqⁿ} is the Fq-subspace U_C^⊥ ={(h⁰₀(x), . . . , h⁰_n−r−1(x)) :x∈Fqⁿ}.

Proof. By Remark 4.11, up to equivalence,C=hh₀(x), . . . , hr−1(x)i_Fqn, for someh₀(x), . . . , hr−1(x) as in (13), andC^⊥=hh⁰₀(x), . . . , h⁰_n−r−1(x)i_Fqn, for someh⁰₀(x), . . . , h⁰_n−r−1(x) as in (14). Note that, since C is an MRD-code, the linearized polynomialsh0(x), . . . , hr−1(x) have no common roots other than 0 since otherwise the code would not contain invertible maps, see e.g.

[22, Lemma 2.1]. Our aim is to show that applying the duality introduced in Section 3 toUC ={(h₀(x), . . . , hr−1(x)) :x ∈ Fqⁿ} we get the Fq-subspace U_C^⊥ ={(h⁰₀(x), . . . , h⁰_n−r−1(x)) :x ∈ Fqⁿ}. By Result 4.7 we have that UC

is a maximum (r−1)-scattered Fq-subspace of F^r_qⁿ. If n > r, i.e. C has minimum distance greater than one, we can embed Λ =hU_Ci_Fqn in Fⁿqⁿ in such a way that

Λ =

(x₀, x₁, . . . , x_r, . . . , xn−1)∈Fⁿ_qⁿ :x_j = 0 j /∈ {t₀, . . . , tr−1} , and hence the vector (h0(x), . . . , hr−1(x)) of UC is extended to the vector (a₀, a₁, . . . , an−1) ofFⁿqⁿ as follows

ai =

h_i(x) ifi∈ {t₀, . . . , tr−1}, 0 otherwise.

Let Γ be the Fqⁿ-subspace of Fⁿqⁿ of dimension n−r represented by the equations

Γ :















xt0 =− X

j /∈{t0,...,tr−1}

g0,jxj

...

x_tr−1 =− X

j /∈{t₀,...,tr−1}

gr−1,jx_j

and letW ={(x, x^q, . . . , x^qn−1) :x∈Fqⁿ}. It can be seen that Γ∩W ={0}, otherwise the polynomials h₀(x), . . . , hr−1(x) would have a common root.

Also

UC=hW,Γi_Fq ∩Λ.

Let β:Fⁿqⁿ ×Fⁿqⁿ → Fqⁿ be the standard inner product, i.e. β((x,y)) = Pn−1

i=0 x_iy_i where x = (x₀, . . . , xn−1) and y = (y₀, . . . , yn−1). Also, the re- striction ofβ overW ×W isβ

W×W((x, x^q, . . . , x^qn−1),(y, y^q, . . . , y^qn−1)) = Tr_qⁿ_/q(xy). Furthermore, with respect to the orthogonal complement oper- ation ⊥defined byβ we have that

Γ^⊥:xj =

r−1

X

`=0

g_{j,`</sub>xt<sub>`} j /∈ {t₀, . . . , tr−1}.

Then the Delsarte dual ¯UC ofUC is theFq-subspace W+ Γ^⊥ of the quotient space Fⁿqⁿ/Γ^⊥ isomorphic to U⁰ := hW,Γ^⊥i_Fq ∩Λ⁰, where Λ⁰ is the Fqⁿ- subspace ofFⁿ_qⁿ of dimensionn−r represented by the following equations

Λ⁰ :xt0 =. . .=xtr−1 = 0.

By identifying Λ⁰ with F^n−rqⁿ , direct computations show thatU⁰ can be seen as the Fq-subspace U_C^⊥ ={(h⁰₀(x), . . . , h⁰_n−r−1(x)) : x∈ Fqⁿ} of dimension nof F^n−r_qⁿ , i.e. U⁰ =U_C⊥.

5 Intersections of maximum h-scattered subspaces with hyperplanes

This section is devoted to prove

Theorem 2.7If U is a maximumh-scatteredFq-subspace of a vector space V(r, qⁿ) of dimension rn/(h+ 1), then for any (r −1)-dimensional Fqⁿ- subspaceW of V(r, qⁿ) we have

rn

(h+ 1)−n≤dim_Fq(U ∩W)≤ rn

(h+ 1) −n+h.

As we already mentioned, the theorem above is a generalization of [4, Theorem 4.2], which is the h = 1 case of our result. In that paper, the number of hyperplanes meeting a 1-scattered subspace of dimensionrn/2 in a subspace of dimensionrn/2−nor rn/2−n+ 1 has been determined as well. Subsequently to this paper, in [29] (see also [23] for the h = 2 case), such values have been determined for everyh.

5.1 Preliminaries on Gaussian binomial coefficients The Gaussian binomial coefficient

n k

q

is defined as the number of the k- dimensional subspaces of then-dimensional vector space Fⁿ_q. Hence

n k

q

=







1 ifk= 0

(1−qⁿ)(1−qⁿ⁻¹)...(1−q^n−k+1)

(1−q^k)(1−q^k−1)...(1−q) if 1≤k≤n

0 ifk > n.

(15) Recall the following properties of the Gaussian binomial coefficients.

n k

q

k j

q

= n

j

q

n−j k−j

q

, (16)

n k

q

= n

n−k

q

. (17)

n−1

Y

j=0

(1 +q^jt) =

n

X

j=0

q^j(j−1)/2 n

j

q

t^j, (18)

Definition 5.1. The q-Pochhammer symbol is defined as (a;q)_k = (1−a)(1−aq). . .(1−aq^k−1).

Theorem 5.2 (q-binomial theorem [15, pg. 25, Exercise 1.3 (i)]).

(ab;q)_n=

n

X

k=0

b^k n

k

q

(a;q)_k(b;q)n−k, (19)

(ab;q)_n=

n

X

k=0

a^n−k n

k

q

(a;q)_k(b;q)n−k. (20) Corollary 5.3. In (19) and (20) put a=q^−nr/s and b=q^nr/s−n to obtain

(q⁻ⁿ;q)s=

s

X

j=0

q^j(nr/s−n) s

j

q

(q^−nr/s;q)j(q^nr/s−n;q)s−j, (21)

(q⁻ⁿ;q)s=q^−nr

s

X

j=0

q^jnr/s s

j

q

(q^−nr/s;q)j(q^nr/s−n;q)s−j, (22) respectively.

The l-th elementary symmetric function of the variables x1, x2, . . . , xn

is the sum of all distinct monomials which can be formed by multiplying togetherl distinct variables.

Definition 5.4. Denote by σ_k,l the l-th elementary symmetric polynomial in k+ 1 variables evaluated in 1, q, q², . . . , q^k.

Lemma 5.5 ([6, Proposition 6.7 (b)]).

σ_k,l=q^l(l−1)/2 k+ 1

l

q

.

We will also need the followingq-binomial inverse formula of Carlitz.

Theorem 5.6 ([7, special case of Theorem 2, pg. 897 (4.2) and (4.3)]).

Suppose that{a_k}_k≥0 and{b_k}_k≥0 are two sequences of complex numbers. If a_k = Pk

j=0(−1)^jq^j(j−1)/2 k

j

q

b_j, then b_k = Pk

j=0(−1)^jqj(j+1)/2−jk

k j

q

a_j and vice versa.

5.2 Double counting

Puts=h+ 1|rnand letU be anrn/s-dimensionalFq-subspace ofV(r, qⁿ) such that for each (s−1)-dimensionalFqⁿ-subspace W, we have dim_Fq(W∩ U)≤s−1.

Lethi denote the number of (r−1)-dimensionalFqⁿ-subspaces meeting U in an Fq-subspace of dimension i. It is easy to see that

h_i = 0 for i < rn s −n.

In PG(V,Fqⁿ) = PG(r−1, qⁿ), the integer hi coincides with the number of hyperplanes PG(W,Fqⁿ) such that dim_Fq(W ∩U) =i. Also, the number of hyperplanes is (q^rn−1)/(qⁿ−1), which is the same asP

ih_i, thus X

i

h_i(qⁿ−1) =q^rn−1. (23) Fork∈ {0,1, . . . , s−1} we can double count the set

{(H,(P₁, P₂, . . . , P_k+1)) :H is a hyperplane, P₁, P₂, . . . , P_k+1∈H∩L_U and hP₁, P₂, . . . , P_k+1i ∼= PG(k, q)}.

By Proposition 2.1 this gives X

i

h_i

qⁱ−1 q−1

qⁱ−q q−1

. . .

qⁱ−q^k q−1

= q^rn/s−1

q−1

! q^rn/s−q q−1

!

. . . q^rn/s−q^k q−1

! q^(r−k−1)n−1 qⁿ−1

! ,

or equivalently Lemma 5.7.

X

i

hi(qⁿ−1)(qⁱ−1)(qⁱ−q)(qⁱ−q²). . .(qⁱ−q^k) = (q^rn/s−1)(q^rn/s−q)(q^rn/s−q²). . .(q^rn/s−q^k)(q^(r−k−1)n−1).

Our aim is to prove A:=X

i

h_i(qⁿ−1)(qⁱ−q^n(r−s)/s). . .(qⁱ−qn(r−s)/s+s−1) = 0.

This would clearly yieldh_i = 0, fori > n(r−s)/s+s−1, and hence Theorem 2.7.